# Higher Spin Extension of Fefferman-Graham Construction

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## Abstract

**:**

## 1. Introduction

## 2. Off-Shell Fefferman-Graham Theory

## 3. On-Shell Fefferman-Graham Theory

- This system is equivalent to on-shell gravity in $d+1$ dimensions with a nonvanishing cosmological constant (in other words the metric ${g}_{\mu \nu}$ is Einstein). The spacetime manifold can be identified with the curved hyperboloid ${V}^{2}=-1$;
- This system describes conformal gravity in d dimensions. For d odd it is off-shell, while for d even it is on-shell, the field equations resulting from the conformal anomaly (For d even, in the original FG approach the Ricci flatness was imposed only up to a certain power of the defining function so that the conformal gravity was always off-shell. Another point of view is to require Ricci flatness at all orders which results in conformal gravity equations. Note that in this case the system also describes subleading solutions). The spacetime manifold can be identified with the projectivization of the curved hypercone ${V}^{2}=0$;

## 4. Higher-Spin Extension of Fefferman-Graham Theory

#### 4.1. Off-Shell Higher-Spin Fields on Gravitational Backgrounds

#### 4.2. Poisson Bracket vs. Star-Product

## 5. Towards on-Shell Higher-Spin Theory

#### 5.1. Parent Reformulation

#### 5.2. Factorization

#### 5.3. Relation to Unfolded Equations

## 6. Conclusions and Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. FG Ambient Construction as $\mathfrak{sp}(2)$ Algebra of Constraints

#### Appendix A.1. Klein Flat Ambient Model

#### Appendix A.2. Fefferman-Graham Ambient Construction

**Conformal space:**

**Example.**

**Metric bundle:**

**Example.**

**Ambient space:**

- its signature has one more timelike and one more spacelike direction with respect to ${g}_{ab}$,
- it is homogeneous of degree two with respect to the homotheties: ${\mathcal{L}}_{V}{G}_{MN}=2\phantom{\rule{0.166667em}{0ex}}{G}_{MN}$,
- the one-form ${V}_{M}={G}_{MN}{V}^{N}$ is closed.

**Example.**

**Bulk:**

**Example.**

#### Appendix A.3. Properties of the Ambient Metric and of the Homothety Vector Field

- (I)
- The ambient metric is of homogeneity degree two with respect to the homothety vector field:$${\mathcal{L}}_{V}{G}_{AB}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{0.166667em}{0ex}}{G}_{AB}\phantom{\rule{0.166667em}{0ex}}.$$
- (II)
- The homothety one-form is closed:$${\partial}_{[A}{V}_{B]}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}.$$

- (III)
- The ambient metric is equal to the covariant derivative of the homothety one-form:$${G}_{AB}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\nabla}_{A}{V}_{B}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\nabla}_{B}{V}_{A}\phantom{\rule{0.166667em}{0ex}}.$$$${\nabla}_{A}{V}^{B}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\delta}_{A}^{B}\phantom{\rule{0.166667em}{0ex}}.$$

**Proof.**

- (IV)
- The homothety one-form is equal to half the gradient of the homothety vector field squared:$${V}_{A}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\partial}_{A}\left(\frac{{V}^{2}}{2}\right)\phantom{\rule{0.166667em}{0ex}}.$$

**Proof.**

#### Appendix A.4. Hypersurface Orthogonality and Homogeneity as $\mathfrak{sp}(2)$ Algebra

**Proposition.**

- the scalar field is equal to $F(X)=-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{V}^{M}(X)\phantom{\rule{0.166667em}{0ex}}{G}_{MN}(X)\phantom{\rule{0.166667em}{0ex}}{V}^{N}(X)$,
- the symmetric tensor field ${G}_{MN}(X)$ and the vector field ${V}^{M}(X)$ obey the properties (I)-(IV).

**Proof.**

## Appendix B. Covariant Derivatives

## Appendix C. Off-shell vs On-shell, Boundary vs. Bulk

#### Appendix C.1. Off-Shell Boundary Scalar Field

#### Appendix C.2. On-Shell Bulk Scalar Field

#### Appendix C.3. On-Shell Boundary Scalar Field (Aka Singleton)

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Bekaert, X.; Grigoriev, M.; Skvortsov, E.
Higher Spin Extension of Fefferman-Graham Construction. *Universe* **2018**, *4*, 17.
https://doi.org/10.3390/universe4020017

**AMA Style**

Bekaert X, Grigoriev M, Skvortsov E.
Higher Spin Extension of Fefferman-Graham Construction. *Universe*. 2018; 4(2):17.
https://doi.org/10.3390/universe4020017

**Chicago/Turabian Style**

Bekaert, Xavier, Maxim Grigoriev, and Evgeny Skvortsov.
2018. "Higher Spin Extension of Fefferman-Graham Construction" *Universe* 4, no. 2: 17.
https://doi.org/10.3390/universe4020017